Suppose you want to find the minimum and maximum elements of an unsorted array of length n$.$ One way is to scan through the array once to find the maximum $($which requires n $1$ comparisons between values$),$ and scan through it a second time to find the minimum $($which also requires n $2$ comparisons between values if you eliminate the maximum element first$),$ for a total of $2$n $3$ comparisons. You can do this more efficiently with divide$-$and$-$conquer. Design a divide$-$and$-$conquer algorithm to find the minimum and maximum elements of an unsorted array, which performs at most $(3$n$/2)2$ comparisons between elements in total.

$($a$)$ Present a recursive divide$-$and$-$conquer algorithm for solving this problem.

$($b$)$ Give a recurrence relation, including base case$($s$),$ that describes the num$-$ ber of comparisons in each call to your recursive function. Your recurrence relation should be a function of n$,$ where n is the size of the array.

$($c$)$ Prove that the value of your recurrence is at most $(3$n$/2)2$ for all n $>0.$

For simplicity, you can assume that n is a power of $2.$ Make use of the fact that$1+2+4+...+2$p $=2$p$+11$